{
  "schema_version": 1,
  "problem_number": "AMR-011-0025",
  "title": "A Free-Uniform-Spanning-Forest Proof of Finite-Index Multiplicativity",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "An explicit question of Abért asks for a free-spanning-forest proof that the first L²-Betti number is multiplicative under passage to a finite-index subgroup. We give such a proof using the free uniform spanning forest (FUSF). For H ≤ Γ of index k, lift a spanning tree of the finite Schreier quotient to a forest W in a labeled Cayley multigraph G, and contract its finite components to obtain a Cayley multigraph X of H. The closed finite-cycle space of G is the graph of a bounded H-equivariant operator over that of X. Equivariant dimensions therefore give the FUSF intensity identity I_H(G) = (k−1) + I_H(X). Combining this with the FUSF formula I_K = 1 + β₁⁽²⁾(K) yields β₁⁽²⁾(H) = kβ₁⁽²⁾(Γ) without importing finite-index multiplicativity of von Neumann dimension.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.GR",
    "math.PR"
  ],
  "keywords": [
    "L²-Betti numbers",
    "uniform spanning forests",
    "finite-index subgroups",
    "Cayley graphs",
    "von Neumann dimension",
    "L2-Betti number",
    "free uniform spanning forest",
    "finite-index subgroup",
    "Schreier graph",
    "AMR-011-0025",
    "open mathematics",
    "mathematical proof",
    "math.GR",
    "math.PR"
  ],
  "manuscript_version_date": "2026-09-01",
  "publication_date": "2026-09-02",
  "publication_date_kind": "first public online release",
  "version": "1.0 (typesetting revision 2026-09-05)",
  "date_modified": "2026-09-05",
  "presentation_revision_only": true,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/amr-011-0025/",
  "pdf_url": "https://eulersolve.org/papers/amr-011-0025/paper.pdf?v=2c2230667cf2",
  "doi": "10.5281/zenodo.22245583",
  "zenodo_record_url": "https://zenodo.org/records/22245583",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "The finite-index multiplicativity example is fully proved; no claim is made that every request in the source list is settled.",
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    "verification_report.md": {
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  "ai_use_disclosure": "AI-assisted tools supported literature search, computation, proof auditing, and manuscript preparation. The author remains responsible for all claims and the final text."
}